Problem 34 - Elementary and Intermediate Algebra

Question

Factor. \(a^{2}-(b-c)^{2}\)

Step-by-Step Solution

Answer

The expression factors to \((a - b + c)(a + b - c)\).

Step 1: Recognize the Formula

Identify the expression as a difference of squares. A difference of squares has the form \(x^2 - y^2\), which can always be factored into \((x - y)(x + y)\). Here, \(a^2 - (b-c)^2\) fits the pattern where \(x = a\) and \(y = b-c\).

Step 2: Apply the Difference of Squares Formula

Substitute \(a\) and \(b-c\) into the formula for factoring the difference of squares. This gives us: \((a - (b-c))(a + (b-c))\).

Step 3: Simplify the Expression

Simplify each part of the factored expression. Start with \(a - (b-c)\), which simplifies to \(a - b + c\), and \(a + (b-c)\), which simplifies to \(a + b - c\). Therefore, the factored form of \(a^2 - (b-c)^2\) is \((a - b + c)(a + b - c)\).

Key Concepts

Factoring Expressions

Factoring expressions is a fundamental skill in algebra that allows you to rewrite expressions in a simpler or more useful form. It involves breaking down a complex expression into the product of two or more simpler expressions. This is especially useful for solving equations or simplifying complex algebraic expressions.

• Imagine you are working with numbers and trying to find the factors of 12. You can break it down into 3 and 4, or 2, 2, and 3.
• Similarly, in algebra, factoring turns a polynomial like \(a^2 - (b-c)^2\) into a product such as \((a - b + c)(a + b - c)\).

Understanding how to factor expressions expands your toolbox, making complex math problems more manageable. You'll often find yourself using skills like recognizing patterns and applying known formulas, such as the difference of squares, to successfully factor algebraic expressions. By practicing these techniques, factoring becomes second nature and aids in tackling more advanced algebraic concepts.

Algebraic Identities

Algebraic identities are proven equations that hold true for all values of the variables involved. Think of them as universal truths in mathematics that always work regardless of the numbers you plug in. The difference of squares is a classic example, with its formula \(x^2 - y^2 = (x - y)(x + y)\).

• These identities give us shortcuts to simplify and manipulate algebraic expressions quickly.
• For example, recognizing that \(a^2 - (b-c)^2\) is a difference of squares enables you to factor it swiftly using the identity \((a - (b-c))(a + (b-c))\).

Once you're familiar with algebraic identities, you can effortlessly move through factorization problems. Mastery of identities allows for greater flexibility and speed in mathematical calculations, which is crucial for more advanced studies.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves making an expression easier to read or solve by combining like terms, reducing fractions, or factoring, among other methods. It makes complex problems more straightforward and manageable, which is particularly useful in equation solving and calculus.

• In the given example \((a-(b-c))(a+(b-c))\), simplification helps us view the expression more clearly.
• Simplifying \(a - (b-c)\) gives \(a - b + c\), and \(a + (b-c)\) gives \(a + b - c\).

This step of simplification is crucial to ensure you have the simplest form possible before using it in further calculations. It eliminates unnecessary complexity and helps in understanding the structure and pattern of the expression at a deeper level. Without simplification, solving algebraic problems would become a cumbersome process with more room for error.